Theorem elirr 8045

Description: No class is a member of itself. Exercise 6 of [TakeutiZaring] p. 22. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)

Assertion

Ref	Expression
elirr

Proof of Theorem elirr

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 22	. . . . 5
2	1, 1	eleq12d 2539	. . . 4
3	2	notbid 294	. . 3
4		elirrv 8044	. . 3
5	3, 4	vtoclg 3167	. 2
6		pm2.01 168	. 2
7	5, 6	ax-mp 5	1

Syntax hints:  -.wn 3  ->wi 4  =wceq 1395  e.wcel 1818

This theorem is referenced by:  sucprcreg  8046  sucprcregOLD  8047  alephval3  8512  rankeq1o  29828  hfninf  29843  bnj521  33792  bj-disjcsn  34505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-reg 8039

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-nul 3785  df-sn 4030  df-pr 4032